Warp image to fit UV unwrap, without using projection unwrapping
In the end, I resulted to keep the same texture as it is and not warp it. Then I experimented around a bit until I found my solution.


First of all, I increased the resolution from 900px to 2048px - this will allow more pixels on steep gradients, reducing the stretch.
Next, when creating my normal map, I set the blur to -10 to make the normal have a gradual change, to create more of a rounded impression than a square-shaped dent.


Following these steps will get you the desired result.